[A04] Soundness

It should be obvious by now that validity is about the logical connection between the premises and the conclusion. When we are told that an argument is valid, this is not enough to tell us anything about the actual truth or falsity of the premises or the conclusion. All we know is that there is a logical connection between them, that the premises entail the conclusion.

So even if we are given a valid argument, we still need to be careful before accepting the conclusion, since a valid argument might contain a false conclusion. What we need to check further is of course whether the premises are true. If an argument is valid, and all the premises are true, then it is called a sound argument. Of course, it follows from such a definition that a sound argument must also have a true conclusion. In a valid argument, if the premises are true, then the conclusion cannot be false, since by definition it is impossible for a valid argument to have true premises and a false conclusion in the same situation. So given that a sound argument is valid and has true premises, its conclusion must also be true. So if you have determined that an argument is indeed sound, you can certainly accept the conclusion.

An argument that is not sound is an unsound argument. If an argument is unsound, it might be that it is invalid, or maybe it has at least one false premise, or both.

Is it possible to have arguments of the following kinds? If so, provide an example. If not, explain why. It is particularly important to note the highlighted cases.

  True conclusion
true premises
True conclusion
false premises
False conclusion
true premises
False conclusion
false premises
Valid & sound
Valid & unsound
Invalid & sound
Invalid & unsound

Are the following statements true or false? Why?

  1. All invalid arguments are unsound.
  2. All true statements are valid.
  3. To show that an argument is unsound, we must at least show that some of its premises are actually false.
  4. An invalid argument must have a false conclusion.
  5. If all the premises of a valid argument are false, then the conclusion must also be false.
  6. If all the premises and the conclusion of an argument are true, then the argument is valid.
  7. All sound arguments are true.
  8. Any valid argument with a true conclusion is sound.

Here are some common questions from students about validity and soundness and their answers. See if you know how to answer them yourself.

  1. An argument is valid, if when all the premises are true the conclusion is true. What if the premises are inconsistent? What if it is impossible for all the premises to be true at the same time? Is the argument still valid?
  2. If the conclusion of an argument is tautological, does that means that the argument is valid?
  3. If every argument with a tautological conclusion is valid, then shouldn't this argument be valid rather than invalid?
  4. (Premise) All cows are mammals.
    (Conclusion) Therefore, the sun is larger than the moon.

    Isn’t the conclusion a tautology?

  5. "It is raining. Therefore 1+1=2." Why is this a valid argument as the premise and the conclusion are talking about completely different things?
  6. I know that all sound arguments must have true conclusions because such arguments are by definition valid with all true premises. On the other hand, if an argument is valid and has a true conclusion, does it follow that it is sound?
  7. "All arguments are either valid or unsound." Is this statement true?
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